Uniqueness of dilation invariant norms

Abstract

Let δ a \delta _a be a nontrivial dilation. We show that every complete norm ‖ ⋅ ‖ \|\cdot \| on L 1 ( R N ) L^1(\mathbb {R}^N) that makes δ a \delta _a from ( L 1 ( R N ) , ‖ ⋅ ‖ ) (L^1(\mathbb {R}^N),\|\cdot \|) into itself continuous is equivalent to ‖ ⋅ ‖ 1 \|\cdot \|_1 . δ a \delta _a also determines the norm of both C 0 ( R N ) C_0(\mathbb {R}^N) and L p ( R N ) L^p(\mathbb {R}^N) with 1 > p > ∞ 1>p>\infty in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on L ∞ ( R N ) L^\infty (\mathbb {R}^N) .